Now there is a post process script available, but I think it will only work as long as Filebot is installed on the same "machine"? Am I right?
Or is there any way to trigger Filebot from within NZBGet on an IP or something like this?

If not, is there a way to install FIlebot (+ Filebot Node) to Ubuntu? All I've seen seems to be using the GUI of Ubuntu, which I don't have.

The clean option is to have NZBGet call a command, and this command would then ssh into the QNAP NAS and call the filebot command on the NAS.

Alternatively, if you must use HTTP requests, and if you have FileBot Node running, then you can using the Request Inspector in your browser to see what HTTP requests are sent (i.e. parameters, headers, etc) when you click Execute in FileBot Node, and then you can just replay that request.